## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### On log surfaces

#### Abstract

This paper is an announcement of the minimal model theory for log surfaces in all characteristics and contains some related results including a simplified proof of the Artin–Keel contraction theorem in the surface case.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci. Volume 88, Number 8 (2012), 109-114.

Dates
First available: 4 October 2012

http://projecteuclid.org/euclid.pja/1349355140

Digital Object Identifier
doi:10.3792/pjaa.88.109

Zentralblatt MATH identifier
06126092

Mathematical Reviews number (MathSciNet)
MR2989060

Subjects
Primary: 14E30: Minimal model program (Mori theory, extremal rays)
Secondary: 14D06: Fibrations, degenerations

#### Citation

Fujino, Osamu; Tanaka, Hiromu. On log surfaces. Proceedings of the Japan Academy, Series A, Mathematical Sciences 88 (2012), no. 8, 109--114. doi:10.3792/pjaa.88.109. http://projecteuclid.org/euclid.pja/1349355140.

#### References

• M. Artin, Some numerical criteria for contractability of curves on algebraic surfaces, Amer. J. Math. 84 (1962), 485–496.
• L. Bădescu, Algebraic surfaces, translated from the 1981 Romanian original by Vladimir Maşek and revised by the author, Universitext, Springer, New York, 2001.
• P. Cascini, J. M$^{\text{c}}$Kernan and M. Mustaţă, The augmented base locus in positive characteristic. (Preprint).
• G. Frey and M. Jarden, Approximation theory and the rank of abelian varieties over large algebraic fields, Proc. London Math. Soc. (3) 28 (1974), 112–128.
• O. Fujino, Minimal model theory for log surfaces, Publ. Res. Inst. Math. Sci. 48 (2012), no. 2, 339–371.
• T. Fujita, Vanishing theorems for semipositive line bundles, in Algebraic geometry (Tokyo/Kyoto, 1982), 519–528, Lecture Notes in Math., 1016, Springer, Berlin, 1983.
• T. Fujita, Semipositive line bundles, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1983), no. 2, 353–378.
• T. Fujita, Fractionally logarithmic canonical rings of algebraic surfaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1984), no. 3, 685–696.
• S. Keel, Basepoint freeness for nef and big line bundles in positive characteristic, Ann. of Math. (2) 149 (1999), no. 1, 253–286.
• D. Knutson, Algebraic spaces, Lecture Notes in Math., 203, Springer, Berlin, 1971.
• J. Kollár, Singularities of the minimal model program. (Preprint).
• J. Kollár and S. Kovács, Birational geometry of log surfaces. (Preprint).
• J. Kollár and S. Mori, Birational geometry of algebraic varieties, translated from the 1998 Japanese original, Cambridge Tracts in Math., 134, Cambridge Univ. Press, Cambridge, 1998.
• V. Maşek, Kodaira-Iitaka and numerical dimensions of algebraic surfaces over the algebraic closure of a finite field, Rev. Roumaine Math. Pures Appl. 38 (1993), no. 7–8, 679–685.
• H. Tanaka, Minimal models and abundance for positive characteristic log surfaces. (Preprint).
• H. Tanaka, X-method for klt surfaces in positive characteristic. (Preprint).
• B. Totaro, Moving codimension-one subvarieties over finite fields, Amer. J. Math. 131 (2009), no. 6, 1815–1833.